The paper solves graph realization problems for Reeb graphs of Morse functions.
problem Realizing graphs as Reeb graphs with specific preimage configurations.
method Constructing Morse functions with prescribed preimages.
result Solved realization problems for certain types of graphs.
New method realizes planar graphs as Reeb graphs of algebraic functions.
problem Realizing planar graphs as Reeb graphs of algebraic functions.
method Generic embedding and elementary procedures.
result Generically embedded planar graphs are homeomorphic to Reeb graphs of algebraic functions.
Framework for universal graph function approximators outperforms existing methods.
problem Graph classification and separation of graph classes.
method Inspired by persistent homology, dependency parsing, and multivalued functions, the framework constructs universal approximators on graph isomorphism classes.
result Achieves state-of-the-art performance on four graph datasets.
We consider adaptations of the Mumford-Shah functional to graphs. These are based on discretizations of nonlocal approximations to the Mumford-Shah functional. Motivated by applications in machine learning we study the random geometric graphs associated to random samples of a measure. We establish the conditions on the…
Study of digital topology concepts like hyperspaces and function graphs.
problem Adapting classical topology concepts to digital topology.
method Define digital hyperspaces and function graphs, study their properties.
result Some relationships and graphical properties of digital hyperspaces and function graphs.
Graph Neural Nets (GNNs) have received increasing attentions, partially due to their superior performance in many node and graph classification tasks. However, there is a lack of understanding on what they are learning and how sophisticated the learned graph functions are. In this work, we propose a dissection of GNNs …
We study the Bakry-Émery curvature function KG,x:(0,∞]→R of a vertex x in a locally finite graph G systematically. Here KG,x(N) is defined as the optimal curvature lower bound K in the Bakry-Émery curvature-dimension inequality $CD(\mathcal{K},\ma…
New graph learning model can approximate any function and handle edge values.
problem Graph learning models' limitations in approximating functions and handling edge values.
method Proposes a Graph Neural Network that can approximate any function and handle arbitrary edge values.
result Proves the model is strictly more expressive than existing models.
We investigate properties that intuitively ought to be satisfied by graph clustering quality functions, that is, functions that assign a score to a clustering of a graph. Graph clustering, also known as network community detection, is often performed by optimizing such a function. Two axioms tailored for graph clusteri…
FuDGE estimates differences between functional graphs in high-dimensional settings.
problem Estimating differences between two undirected functional graphical models with shared structures.
method FuDGE: A method that directly estimates the functional differential graph without first estimating individual graphs.
result FuDGE consistently estimates the functional differential graph in high-dimensional settings.
Bayesian optimisation framework for graph functions.
problem Optimizing functions on graph structures efficiently.
method Learning suitable kernels and local modelling.
result Demonstrated effectiveness on synthetic and real-world graphs.
This paper explores different graph neural network functions to improve graph isomorphism.
problem Lack of robust implementation for graph neural networks due to limited analysis of underlying functions.
method Examines various alternative functions for different modules in GNNs using benchmark datasets.
result Generally used underlying techniques do not always capture the overall graph structure.
Study of harmonic functions on infinite penny graphs.
problem Characterizing harmonic functions on infinite penny graphs.
method Proving volume doubling and Poincaré inequalities, analyzing polynomial growth harmonic functions.
result Finite dimensional property of ancient solutions of the heat equation.
Extends graph similarity theory to improve MPNNs' generalization abilities.
problem Understanding MPNNs' generalization beyond training data.
method Extends graph similarity theory, assesses graph structure, aggregation, and loss functions.
result Improves understanding of MPNNs' generalization properties.
We investigate the problem of the realization of a given graph as the Reeb graph R(f) of a smooth function f:M→R with finitely many critical points, where M is a closed manifold. We show that for any n≥2 and any graph Γ admitting the so called good orientation there exis…
Finite graphs with specific curvature have limited harmonic functions and ends.
problem Graphs with nonnegative curvature outside a finite subset.
method Introducing discrete Gromov-Hausdorff convergence to study bounded harmonic functions.
result The space of bounded harmonic functions is finite dimensional, and the number of non-parabolic ends is finite.
Proposes a novel graph learning framework for robust graph topology learning from graph signals.
problem Graph learning for revealing node relationships in data entities.
method Functional learning with smoothness-promoting graph learning, incorporating Kronecker product kernel.
result Improves robustness against missing and incomplete information in graph signals.
Study polynomial growth harmonic functions on infinite penny graphs.
problem Finite-dimensional property of polynomial growth harmonic functions on infinite penny graphs.
method Asymptotically sharp dimensional estimate for ancient solutions of the heat equation.
result Proved the asymptotically sharp dimensional estimate.
Graph neural networks (GNNs) have been shown to replicate convolutional neural networks' (CNNs) superior performance in many problems involving graphs. By replacing regular convolutions with linear shift-invariant graph filters (LSI-GFs), GNNs take into account the (irregular) structure of the graph and provide meaning…
Proposes a method to adapt labels on graphs with few labeled nodes.
problem Domain adaptation for graphs with limited labeled nodes.
method Optimization problem solving label transfer using spectral graph wavelets.
result Method yields satisfactory classification accuracy compared to existing methods.
New equivariant filters improve graph classification.
problem Designing deep learning models for graph symmetries.
method Nonlinear spectral filters (NLSFs) that are equivariant to graph functional shifts.
result NLSFs outperform existing spectral GNNs in graph classification.
Study Morse functions on projective plane using Reeb graphs.
problem Investigate topological structure of Morse functions on projective plane.
method Use Reeb graphs to describe and prove properties of simple Morse functions on RP2. result Prove that Reeb graphs are a complete topological invariant for simple Morse functions on RP2. Bayesian Optimization for graph node subset functions.
problem Optimizing functions over node subsets in graphs.
method Bayesian Optimization framework for combinatorial optimization on graphs.
result Effectiveness of the proposed BO framework on various graph types and tasks.
The paper extends previous work on Reeb graphs of smooth functions on 3D manifolds to non-orientable cases.
problem Extending the understanding of Reeb graphs to non-orientable 3D manifolds.
method Constructing smooth functions on non-orientable 3D manifolds whose Reeb graphs match prescribed graphs.
result Explicit construction of smooth functions on non-orientable 3D manifolds with prescribed Reeb graphs.
We study the Ollivier-Ricci curvature of graphs as a function of the chosen idleness. We show that this idleness function is concave and piecewise linear with at most 3 linear parts, with at most 2 linear parts in the case of a regular graph. We then apply our result to show that the idleness function of the Cartes…
The sinh-Gordon equation is solved on finite, symmetric graphs.
problem Solving the sinh-Gordon equation with nonzero prescribed functions on finite graphs.
method Uniform a priori estimate to define topological degree, case-by-case calculation of degree, classical sinh-Gordon equation analysis.
result The classical sinh-Gordon equation with nonzero prescribed function is always solvable on finite, symmetric graphs.
New method constructs smooth functions with specific Reeb graphs and preimages on 3D manifolds.
problem Construct smooth functions with prescribed Reeb graphs and preimages on 3D closed manifolds.
method Develops a new approach to realize graphs as Reeb graphs of smooth functions on 3D closed manifolds.
result Provides a best possible solution for functions on 3D closed manifolds.
A common assumption in semi-supervised learning with graph models is that the class label function varies smoothly on the data graph, resulting in the rather strict prior that the label function has low-frequency content. Meanwhile, in many classification problems, the label function may vary abruptly in certain graph …
The Reeb graph of a function on a smooth manifold is the graph obtained as the space of all connected components of level sets such that the set of all vertices coincides with the set of all connected components of level sets including singular points. Reeb graphs are fundamental and important in the algebraic and diff…
We propose a representation of graph as a functional object derived from the power iteration of the underlying adjacency matrix. The proposed functional representation is a graph invariant, i.e., the functional remains unchanged under any reordering of the vertices. This property eliminates the difficulty of handling e…
Proposes a new method to describe graph vertex features using characteristic functions.
problem Describing the distribution of vertex features at multiple scales on graphs.
method Introduces FEATHER, a computationally efficient algorithm to calculate characteristic functions based on random walk transition probabilities.
result Demonstrates that the proposed method creates high-quality graph representations and is robust to data corruption.
The article uses PageRank and persistent homology for scalable graph comparison.
problem Comparing the similarities between complex networks.
method Combines PageRank and persistent homology to compute a scalable graph descriptor.
result Shows the effectiveness of the method on shape mesh datasets.
For a smooth function on a smooth manifold of a suitable class, the space of all connected components of preimages is the graph and called the {\it Reeb graph}. Reeb graphs are fundamental tools in the algebraic and differential topological theory of Morse functions and more general functions which are not so wild. In …
The study proves unique harmonic functions and combinatorial properties of vertex-transitive graphs.
problem Proving combinatorial properties of vertex-transitive graphs.
method Using harmonic functions and quasi-isometry to R, proving uniqueness and combinatorial results. result Connective constant of non-degenerate vertex-transitive graphs is at least the golden mean.
Study of special subgroups of automorphism groups of Kronrod-Reeb graphs for Morse functions on 2-torus.
problem Characterizing subgroups of automorphism groups of Kronrod-Reeb graphs.
method Analysis of diffeomorphisms preserving Morse functions on 2-torus.
result Full description of special classes of automorphism groups.
Graph kernels assess graph similarity for various applications.
problem Assessing similarity between graphs for predictions.
method Review and comparison of existing graph kernels.
result State-of-the-art graph kernels reviewed and compared.
DBGDGM models dynamic brain graphs for better understanding brain function.
problem Previous brain graph models ignore temporal dynamics, limiting their usefulness.
method DBGDGM clusters brain regions into evolving communities and learns dynamic node embeddings.
result DBGDGM outperforms baselines in graph generation, dynamic link prediction, and graph classification.
Optimal Reeb graphs identified for polygon decomposition.
problem Investigating the topological structure of planar polygon decomposition.
method Using oriented Reeb graphs with a marked vertex for height functions.
result Described all possible optimal Reeb graphs for specific polygon configurations.
We consider immersions admitting uniform graph representations over the affine tangent space over a ball of fixed radius r>0. We show that for sufficiently small C^0-norm of the graph functions, each graph function is smooth with small C^1-norm.
The paper extends decay estimates to graphs with positive spectrum.
problem Proving decay estimates for nonnegative functions on graphs.
method Sharp ℓ2 decay estimates for nonnegative generalized subharmonic functions. result Extends Li and Wang's result to graphs with positive Laplacian spectrum.
Graph neural Thompson Sampling improves online decision-making for graph data.
problem Online decision-making with graph-structured rewards.
method GNN-TS algorithm using GNN for mean reward estimation and graph neural tangent features for uncertainty.
result GNN-TS achieves a state-of-the-art regret bound of ildeO((ildedT)1/2). We consider the setting of Reeb graphs of piecewise linear functions and study distances between them that are stable, meaning that functions which are similar in the supremum norm ought to have similar Reeb graphs. We define an edit distance for Reeb graphs and prove that it is stable and universal, meaning that it pr…
We give a closed formula for the multivariable Conway potential function of any graph link in a homology sphere. As corollaries, we answer three questions by Walter Neumann about graph links.
We introduce a novel encoder-decoder architecture to embed functional processes into latent vector spaces. This embedding can then be decoded to sample the encoded functions over any arbitrary domain. This autoencoder generalizes the recently introduced Conditional Neural Process (CNP) model of random processes. Our ar…
Proves existence and uniqueness of Killing graphs with prescribed curvature.
problem Existence and uniqueness of Killing graphs with prescribed curvature.
method Proves existence and uniqueness of Killing graphs with prescribed mean curvature considering non-constant functions.
result Existence and uniqueness of Killing graphs with prescribed curvature.
e-GGPs learn graph vertex transitions over time.
problem Static graph Gaussian Processes cannot handle dynamic graph structures.
method Proposes e-GGPs with a transition function and neighbourhood kernel.
result e-GGPs outperform static GGPs on time-series regression.
Researchers reconstruct algebraic maps onto curves based on prescribed Reeb graphs.
problem Reconstructing smooth real algebraic maps onto curves with specific Reeb graphs.
method Developed a method to reconstruct functions from general finite graphs, focusing on curves.
result Reconstructed functions from prescribed Reeb graphs, providing a new approach in real algebraic geometry.
Knowledge graphs are a versatile framework to encode richly structured data relationships, but it can be challenging to combine these graphs with unstructured data. Methods for retrofitting pre-trained entity representations to the structure of a knowledge graph typically assume that entities are embedded in a connecte…